$\"\"$

The function is $\"f(x)=\\frac{1}{(1+\\sec$ .

The function is in the form of composite function $\"h\\$ .

Here $\"g(x)=\\sec$ and $\"\\small$ .

Consider $\"g(x)=\\sec$ .

Differentiate on each side with respect to $\"x\"$ .

$\"\\small$

$\"\\small$ .

Consider $\"\\small$ .

Differentiate on each side with respect to $\"x\"$ .

$\"\\\\h\'(x)=-2(1+x)^{-2-1}\\\\$

$\"\"$

$\"f(x)=\\frac{1}{(1+\\sec$ .

Chain Rule of derivatives: $\"\\frac{d}{dx}h(g(x))=$ .

Substitute $\"\\\\h\'(x)=\\frac{-2}{(1+x)^{3}}\"$ and $\"\\small$ in chain rule.

$\"f\'(x)=\\frac{-2}{(1+\\sec$

$\"f\'(x)=\\frac{-2\\sec$

$\"\"$

$\"f\'(x)=\\frac{-2\\sec$